Evaluate the following limit: Evaluate: _ n 1.2+2.3+3.4+ +n(n+1) … - Vidyadip

Question

Evaluate the following limit:
Evaluate: $\lim\limits_{\text{n}\rightarrow\infty}\frac{1.2+2.3+3.4+\ \cdots+\text{n}(\text{n}+1)}{\text{n}^3}$

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Answer

$\lim\limits_{\text{n}\rightarrow\infty}\frac{1.2+2.3+3.4+\ \cdots+\text{n}(\text{n}+1)}{\text{n}^3}$
$=\ \lim\limits_{\text{n}\rightarrow\infty}\frac{\frac{\text{n}(\text{n}+1)(2\text{n}+1)}{6}+\frac{\text{n}(\text{n}+1)}{2}}{\text{n}^3}$
$=\lim\limits_{\text{n}\rightarrow\infty}\frac{\text{n}(\text{n}+1)\Big[\frac{(2\text{n}+1)+3}{6}\Big]}{\text{n}^3}$
$=\lim\limits_{\text{n}\rightarrow\infty}\frac{\frac{\text{n}(\text{n}+1)(2\text{n}+4)}{6}}{\text{n}^3}$
$=\lim\limits_{\text{n}\rightarrow\infty}\frac{\Big(1+\frac{1}{\text{n}}\Big)\Big(2+\frac{4}{\text{n}}\Big)}{6}$
$=\frac{1\times2}{6}$
$=\frac{1}{3}$

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